Mark the Correct alternative in the following:

A student was asked to prove a statement P(n) by induction. He proved P (k + 1) is true whenever P(k) is true for all k > 5 ∈ N and also P(5) is true. On the basis of this he could conclude that P(n) is true.

Mark the Correct alternative in the following:

For all n ∈ N, 3 × 5^{2n + 1} + 2^{3n + 1} is divisible by

Mark the Correct alternative in the following:

If 10ⁿ + 3 × 4^{n + 2} + λ is divisible by 9 for all n ∈ N, then the least positive integral value of λ is

Mark the Correct alternative in the following:

Let P (n): 2n < (1 × 2 × 3 × … × n). Then the smallest positive integer for which P(n) is true is

Mark the Correct alternative in the following:

If P(n) : 49ⁿ + 16ⁿ + λ is divisible by 64 for n ∈ N is true, then the least negative integral value of λ is